- Philip Griffiths Professor of Mathematics
- Professor in the Department of Mathematics
Tuesdays and Thursdays, 10:30-12:00PM, and by appointment
Research Areas and Keywords
Algebra & Combinatorics
Geometry: Differential & Algebraic
PDE & Dynamical Systems
My research concerns problems in the geometric theory of partial differential equations. More specifically, I work on conservation laws for PDE, Finsler geometry, projective geometry, and Riemannian geometry, including calibrations and the theory of holonomy.
Much of my work involves or develops techniques for studying systems of partial differential equations that arise in geometric problems. Because of their built-in invariance properties, these systems often have special features that make them difficult to treat by the standard tools of analysis, and so my approach uses ideas and techniques from the theory of exterior differential systems, a collection of tools for analyzing such PDE systems that treats them in a coordinate-free way, focusing instead on their properties that are invariant under diffeomorphism or other transformations.
I’m particularly interested in geometric structures constrained by natural conditions, such as Riemannian manifolds whose curvature tensor satisfies some identity or that supports some additional geometric structure, such as a parallel differential form or other geometric structures that satisfy some partial integrability conditions and in constructing examples of such geometric structures, such as Finsler metrics with constant flag curvature.
I am also the Director of the Simons Collaboration Special Holonomy in Geometry, Analysis, and Physics, and a considerable focus of my research and that of my students is directed towards problems in this area.
Berger, E, and Griffiths, P. "The Gauss equations and rigidity of isometric embeddings." Duke Math. J. 50.3 (1983): 803-892.
Griffiths, P, and Yang, D. "Characteristics and existence of isometric embeddings." Duke Math. J. 50.4 (1983): 893-994.
Bryant, RL. "Conformal and minimal immersions of compact surfaces into the 4-sphere." J. Differential Geom. 17.3 (1982): 455-473. (Academic Article)
Bryant, RL. "Submanifolds and special structures on the octonians." J. Differential Geom. 17.2 (1982): 185-232.
Bryant, RL. "Holomorphic curves in Lorentzian CR-manifolds." Trans. Amer. Math. Soc. 272.1 (1982): 203-221.
Berger, E, and Griffiths, P. "Some isometric embedding and rigidity results for Riemannian manifolds." Proc. Nat. Acad. Sci. U.S.A. 78.8 (1981): 4657-4660.
Bryant, RL. "Levi-flat Minimal Hypersurfaces in Two-dimensional Complex Space Forms." Adv. Stud. Pure Math., 37, Math. Soc. Japan, Tokyo, 2002, 1--44. Open Access Copy
Bryant, RL, Dunajski, M, and Eastwood, M. "Metrisability of two-dimensional projective structures." Open Access Copy